I also listen to my eBooks and audio books through my iPhone. In my opinion, there is no better app than SMARTER PLAYER. It is easy to use and FREE! However, you must import the audio books from iTunes which I see you don't want to deal with or can't deal with. So, below are other apps I have used. There are many others but I personally find most audio books to be narrated slow so all of the apps below have the option to adjust the speed of narrator's voice (to speed it up or slow it down).
-CloudBeats: Can import audio books directly from your Dropbox. I don’t listen directly from DropBox because the speed is too slow so I will use this app.
-Song Speed: Great app yet it stops playing if you attempt to do anything else on your iPhone (you cannot multitask). I believe it also can access books from DropBox.
Here are a few more iPhone apps that play ABs. I haven’t personally tried any of these out yet:
-AudiPro
-AudiBook
-Capti
-SwiftPlayer
-BookMobile
It took me +1 year to figure out how to use iTunes and once I did, I realized it wasn't as cumbersome as I had made it. If you ever try iTunes again, go for SMARTER PLAYER. I used that for years. Because I am a TTS (text to speech) addict, I no longer use SMARTER PLAYER or any of the apps listed above. Instead, I pay $10/year for Voice Dream because it is by far the best TTS reader for iPhone (I researched for years)! Plus, I can use it to listen to my audio books. If you don't listen to eBooks or use TTS for any other reason, I would not suggest paying for an app like Voice Dream, just mentioning it in case you DO use TTS.
Good luck!